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प्रश्न
Check the commutativity and associativity of the following binary operations '*'. on Q defined by a * b = a − b for all a, b ∈ Q ?
उत्तर
Commutativity:
\[\text{Let } a, b \in Q . \text{Then}, \]
\[ a * b = a - b\]
\[b * a = b - a\]
\[\text{Therefore},\]
\[a * b \neq b * a\]
Thus, * is not commutative on Q.
Associativity :
\[\text{Let }a, b, c \in Q . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b - c \right)\]
\[ = a - \left( b - c \right)\]
\[ = a - b + c\]
\[\left( a * b \right) * c = \left( a - b \right) * c\]
\[ = a - b - c\]
\[\text{Therefore},\]
\[a * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on Q.
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